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Zeno Paradox

  1. Jul 13, 2008 #1
    I may have discovered an explanation to the Zeno Paradox. However, in this explanation, the concept of a race between to objects at different speeds is simplified into one object traveling a given distance. The distance traveled in this example can be represented by the variable D. If the distance traveled were divided into an infinite number of segments, then each segment could be represented by D/(infinite). Therefore, if each of these D/(infinite) segments were added together an infinite amount of times since there is an infinite amount of divisions then if would be D/(infinite) * infinite which would equal D.

    The same is true with time, and as the number of divisions goes to infinite the time to cross those divisions goes to 1/(infinite) then it is easy to resolve the paradox.

    Questions...comments?
     
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  3. Jul 13, 2008 #2

    Hurkyl

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    Then first, you need to precisely state Zeno's paradox. There are many different ways to interpret it. Some interpretations are extremely easy to refute, due to an obvious error in the argument. I've seen others interpret Zeno's paradox as one of the first arguments (by reductio ad absurdum) that that error really is an error. (It probably wasn't so obvious back then)


    What segments?

    What do you mean by (infinite)? And why do you think you can divide by it? And how would the result represent a segment?

    You can't add an infinite number of times. What operation are you really intending to use? (e.g. the basic definition of infinite sum from calculus?)

    What is a division? And why would there be infinitely many of them?

    What does 'infinite' mean here? And why do you think you can multiply D/(infinite) by it?

    And why would you think it gives this result?
     
  4. Jul 13, 2008 #3

    DaveC426913

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    D/infinity*infinity does NOT result in D; it results in undefined - which means 'this has no unique answer'.
     
  5. Jul 13, 2008 #4
    The segments that the line is divided into. If we have a line of length D and we divide it into an infinite amount of sections then each section would be the width of D/(infinite)

    To answer another of your questions right off the bat, (infinite) is equal to infinite just put the parenthesis to be helpful.

    Would you like me to define every word of my post to you?

    A division is: 1. the act or process of dividing; state of being divided.
    Dividing is: 1. to separate into parts, groups, sections, etc.

    There is an infinite amount of divisions because that is the paradox.

    2. indefinitely or exceedingly great: infinite sums of money.

    I can multiply D/(infinite) by infinite because its the same as adding D/(infinite) and infinite amount of times.


    Do me the great honor of learning basic words such as sections and dividing.
     
  6. Jul 13, 2008 #5
    If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.
     
  7. Jul 14, 2008 #6
    Perhaps this can help:

    When dealing with points A and B of finite length, infinite segmentation is classically impossible.
    To go even further, one can not segment infinity, as that would be inherently contradictory. What is one-half of infinity as opposed to infinity itself? (for example)

    Anyway just some thoughts...
     
  8. Jul 14, 2008 #7
    Another point I'd like to offer is that you can keep dividing mathematically forever, but there are practical limits in the real world... even the quantum level.

    When one keeps mathematically dividing below certain levels it simply has no real meaning or effect on reality.
     
  9. Jul 14, 2008 #8

    HallsofIvy

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    All you've really said here is that you do not understand the "paradox". I suspected that when I saw that this had been posted under "General Physics".
     
  10. Jul 14, 2008 #9

    DaveC426913

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    No need to be snooty. And yes, you may need to define terms. That's what Zeno's Paradox is about - our nebulous grasp on meanings versus reality. We find that our words make a lot of assumptions, as you are finding you've done in your argument.
     
  11. Jul 14, 2008 #10

    DaveC426913

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    There is nothing in the paradox about infinity. Infinity only shows up in flawed attempts at a solution.

    The solution to the paradox is to eliminate the infinite. The reason it took so long to be solved is because every other philosopher fell into the same infinity trap that you did.
     
  12. Jul 14, 2008 #11

    rcgldr

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    Isn't infinity a part of Calculus? There are methods used to calculate infinite sums. Integration to find area under a curve is the limit of the area as the number of segments approaches infinity.

    In the case of two objects traveling at different but constant speeds, the ratio of the change in position versus the change in time is constant for each "interval", so the rate of closure and eventually passing by remains constant, even if the limit of the number of intervals approaches infinity while the size of the intervals approaches zero at the point where the faster object passes the slower one.

    Another example, is a ball bouncing with a fixed percentage loss in energy on each bounce. The total time the ball bounces is fixed, but the number of times an abstract ball bounces is infinite. (In real life, eventually the bounces become smaller than the deformation of the ball, so it stops bouncing a bit sooner).
     
  13. Jul 14, 2008 #12

    Hurkyl

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    I would agree that your concept of 'infinite' does not seem to be well-defined. However, mathematics gives it a more than adequate treatment (even to the satisfaction of philosophers!) -- so if you're serious about reflecting about 'infinite' things, you really need to study mathematics.
     
  14. Jul 14, 2008 #13

    DaveC426913

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    Absolutely. And therein will be an answer. But mere arithemetic - dividing and multiplying -i.e. mixing natural numbers and infinities - isn't it.
    That's where the philosphers and the OP went awry.
     
  15. Jul 15, 2008 #14

    rcgldr

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    True, but I'm not sure of the OP's math background. If you replace "infinite" with "n", and take the limit as n approaches infinite, then the OP had the right idea.
     
  16. Jul 15, 2008 #15
    The right idea to prove what? Was Zeno right or wrong in what he was trying to prove?

    The OP seems to think Zeno was claiming any length is equal to an infinite length and any time duration is equal to an infinite duration of time. Or that that is true and thus negates what Zeno was trying to claim.
    I cannot tell because the OP never says what Zeno was claiming to prove.

    But neither does anyone else here which makes me think no one in this argument really knows what Zeno was trying to prove.
     
  17. Jul 15, 2008 #16

    rcgldr

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    This might help:

    http://en.wikipedia.org/wiki/Zeno's_paradoxes

    It's a paradox, which could mean that the author already knew the solution, but still considered it an interesting exercise for a potential reader.
     
  18. Jul 15, 2008 #17
    Infinity/Infinity does not equal one in some cases... true. Such as 5x/x as x-> Infinity. However in this case, the n/n as n-> infinity does equal one.

    Zeno claimed that movement wasn't possible because there was an infinite number of divisions to overcome. What I am saying is that because the number of devisions gets infinitely large, the time to cross those would be infinitely small. So these infinities would cancel.

    If you can't figure out if I am proving or disproving the paradox perhaps it would be helpful to read the Paradox in full from the link Jeff provided. I was mistaken not have added that in my OP.
     
  19. Jul 15, 2008 #18

    DaveC426913

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    Yes.


    No.
     
  20. Jul 16, 2008 #19
    Your assuming, and thus calculating, that infinity resides between two finite points.
     
  21. Jul 16, 2008 #20
    This is not possible.
     
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